Additional resources for A Blichfeldt-type inequality for the surface area

Example text

It then follows that FDint Ftint ∪ (∪α Bα ∩ Ft ) is built from Ftint by attaching n-cells. ht The ‘exterior’ pair (FDext , Ftext ). Let us take a small open Milnor–Lê ball Bj ⊂ Pn × C at every singularity aj ∈ Fb ∩ Sing ∞ f . The ﬁbre Fb is t-regular at inﬁnity at all points X∞ ∩ Fb \ (∪j Bj ), hence it is ϕ-regular at this compact set, since ϕ deﬁnes X∞ . If we denote by Sing (ϕ, f ) the singular locus of the map (ϕ|Cn , f ), then we have the following equality of germs: Sing (ϕ, f ) = (x0 , τ )(i) y in any chart Ui × C and at any point y ∈ X∞ .

The rest of the proof is unchanged. 12 the function d ∞ (x, f (x)) = 1/ρE2 (x), which extends to an analytic function deﬁning X∞ . Another example of a function deﬁning X∞ is the following one, which was and will be used several times. 7 Let h : Pn × K → K be the function which is equal to x0 in (j) (j) (i) every chart Ui × K. A change of chart yields x0 = x0 /xi , which shows that h is K-analytic. This is actually a K-analytic section of the line bundle on Pn × K associated to the divisor H ∞ × K.

4): Sing ∞ τ ⊂ × K. 2 (i) f is a F-type polynomial if its compactiﬁed ﬁbres and their restrictions to the hyperplane at inﬁnity have at most isolated singularities. (ii) f is a B-type polynomial if its compactiﬁed ﬁbres have at most isolated singularities. (iii) f is a W-type polynomial if dim Sing τ ≤ 0. In this case, we say that f has isolated W-singularities at inﬁnity. e. if both its singular set Sing f and the set Sing ∞ f of t-singularities at inﬁnity consist of isolated points. We should stress again that having singularities of the types in the above deﬁnition depends on the chosen system of coordinates x1 , .